Let {a ni , 1 ≤ i ≤ n, n ≥ 1} be an array of real numbers and {X n , n ≥ 1} be a sequence of random variables satisfying the Rosenthal type inequality, which is stochastically dominated by a random variable X. Under mild conditions, we present some results on complete convergence for weighted sums n i=1 a ni X i of random variables satisfying the Rosenthal type inequality. The results obtained in the paper generalize some known ones in the literatures. ∞ n=1 P(|X n − θ| > ε) < ∞.
We study the complete consistency for estimator of nonparametric regression model based onρ~-mixing sequences by using the classical Rosenthal-type inequality and the truncated method. As an application, the complete consistency for the nearest neighbor estimator is obtained.
In this paper, we investigate the parametric component and nonparametric component estimators in a semiparametric regression model based on ϕ-mixing random variables. The r th mean consistency, complete consistency, uniform r th mean consistency and uniform complete consistency are established under some suitable conditions. In addition, a simulation to study the numerical performance of the consistency of the nearest neighbor weight function estimators is provided. The results obtained in the paper improve the conditions in the literature and generalize the existing results of independent random errors to the case of ϕ-mixing random errors.
We study the strong consistency of estimator of fixed design regression model under negatively dependent sequences by using the classical Rosenthal-type inequality and the truncated method. As an application, the strong consistency for the nearest neighbor estimator is obtained.
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